Martingale Inequalities for the Maximum via Pathwise Arguments

TL;DR

This paper develops a new class of martingale inequalities involving the maximum process, derived from pathwise methods, providing bounds on expectations and recovering classical inequalities with new refinements.

Contribution

It introduces a broad class of martingale inequalities based on pathwise arguments, including new refinements for certain p-values and insights into their sharpness.

Findings

Derived upper bounds for expectations of functions of the maximum process

Showed no single inequality is uniformly sharp across all martingales

Recovered and refined Doob's L^p inequalities for p in (0,1]

Abstract

We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry_Labordere et al. (2013) and provide an upper bound on the expectation of a function of the running maximum in terms of marginal distributions at n intermediate time points. The class of inequalities is rich and we show that in general no inequality is uniformly sharp - for any two inequalities we specify martingales such that one or the other inequality is sharper. We then use our inequalities to recover Doob's L^p inequalities. For p in (0,1] we obtain new, or refined, inequalities.